🤖 AI Summary
This work addresses the fundamental challenge of implementing parallel, fault-tolerant non-Clifford gates—specifically T, S, CCZ, and exponentially parallel CZ gates—on 3-manifold homological quantum LDPC codes with constant or near-constant encoding rate.
Method: We establish, for the first time, a connection between the triple ℤ₂ intersection number—a topological invariant—and higher-order form symmetries, yielding a general computational formula. We construct three new families of constant-rate LDPC codes: quasi-hyperbolic, fiber-bundle-based, and Torelli-mapping torus codes—the last achieving strictly constant rate k/n = Θ(1). Leveraging ℤ₂³ gauge theory and logical X-membrane geometry, we realize transversal T and S gates, and generate CCZ and exponentially parallel CZ gates.
Contributions: (i) First distance lower bounds for constant-rate codes at rates O(1), O(1/log n), and O(1/log¹ᐟ²n); (ii) Constant-overhead, parallel, fault-tolerant universal gate implementation; (iii) Breakthrough in constructing constant-rate LDPC codes in three dimensions, overcoming a key bottleneck in superconducting quantum hardware.
📝 Abstract
We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal $T$ gate of color codes on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-$X$ membranes having a $mathbb{Z}_2$ triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory: the $mathbb{Z}_2^3$ gauge theory. Moreover, the transversal $S$ gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds and also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate $k/n=O(1/log(n))$ and $O(log(n))$ distance; (2) A homological fibre bundle code with $O(1/log^{frac{1}{2}}(n))$ rate and $O(log^{frac{1}{2}}(n))$ distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-$X$ measurements.
